The main difficulty in solving the Helmholtz equation within polygons is due to non-analytic vertices. By using a method nearly identical to that used by Fox, Henrici, and Moler in their 1967 paper; it is demonstrated that such eigenvalue calculations can be extended to unprecedented precision, very often to well over a hundred digits, and sometimes to over a thousand digits.A curious observation is that as one increases the number of terms in the eigenfunction expansion, the approximate eigenvalue may be made to alternate above and below the exact eigenvalue. This alternation provides a new method to bound eigenvalues, by inspection.Symmetry must be exploited to simplify the geometry, reduce the number of non-analytic vertices and disentangle degeneracies. The symmetry-reduced polygons considered here have at most one non-analytic vertex from which all edges can be seen. Dirichlet, Neumann, and periodic-type edge conditions, are independently imposed on each polygon edge.The full shapes include the regular polygons and some with re-entrant angles (cut-square, L-shape, 5-point star). Thousand-digit results are obtained for the lowest Dirichlet eigenvalue of the L-shape, and regular pentagon and hexagon.1 Very recently, P. Amore, et al. [1] have independently calculated several highly precise eigenvalues (up to dozens of digits) for various shapes, including the L-shape and the cut-square. They also used the FHM method for some problems, but their focus was on a highorder Richardson extrapolation method with finite elements, which, among other things, can handle more varied shapes. Fortunately, their calculations provide (partial) independent validation of my results.2004, when they demonstrated the numerical advantages of the so-called "generalized singular-value decomposition" (GSVD) method, a relative to the point-matching method.Fortunately, the answer to make the point-matching method work is short and simple: One must both (a) select adequate matching points (both number and distribution) and (b) keep enough precision in the intermediate calculations. My empirical observations are that Chebyshev nodes chosen as matching points (as suggested by Betcke and Trefethen [5]) often work well, even where equally-spaced points do not, and the precision of the intermediate calculations must be significantly higher, often several times, than the precision of the eigenvalue. (a) DC-27 (even and odd) λ ≈ 1044.72 (b) DS-65: λ ≈ 4293.91 (c) DS-1396: λ ≈ 100001.28 FIG. 3. Several representative Dirichlet eigenfunctions within the unit-edged regular pentagon. Blue and yellow areas are opposite sign, and light-dark steps indicate level contours. (a) The degenerate pair just below DS-18 (see Fig. 2d, LEFT). (b) Another example that strongly reveals the pentagram nodal pattern. (c) The first eigenfunction with eigenvalue above 100,000, corresponding to the 8139 th overall eigenvalue, and having about fifty wavelengths along a pentagon edge (Λ ≈ 0.020). Notice how the boundary shape is revealed within the sea of apparent chaos (...
